Before proceeding to discuss possible constraints on
from gravitational
lensing in this section and high redshift supernovae in the next,
let us introduce a quantity which plays a crucial role in these discussions,
namely the luminosity distancedL(z)
up to a given redshift z. Consider an object of absolute luminosity
located at a coordinate
distance r from an observer at r = 0. Light emitted
by the object at a time t is received by the observer at t
= t0,
t and t0 being related by the cosmological
redshift 1 + z = a(t0) /
a(t). The luminosity flux reaching the observer is

The luminosity distance dL depends sensitively
upon both the spatial curvature and the
expansion dynamics of the universe. To demonstrate this we
determine dL using the expression for the coordinate
distance r
obtained by setting ds2 = 0 in (2), resulting in

(24)

which gives

(25)

where =
0tcdt / a(t).

Furthermore, since
dz / dt = - (1 + z)H(z), we get

(26)

where h(z) = H(z) / H0 is
defined in (18), and, in a universe with several components

(27)

Substituting
(27) and (26) in (23) we get the following expression for the luminosity
distance in a multicomponent universe with a cosmological term
[26]

Figure 4. The luminosity distance
dL (in units of H0-1)
is shown as a function of cosmological redshift
z for flat cosmological models with a cosmological constant
m +
= 1.
Heavier lines correspond to larger values of
m. For
comparison we also show (dashed line) the angular size in a flat de Sitter
universe ( = 1).

Before we turn to applications, let us consider a simple
example which
provides us with an insight into the role played by the
luminosity distance dL in cosmology.
In a spatially flat universe the expression for dL
simplifies considerably,
so that we get for the matter dominated model (at2/3):

(30)

On the other hand in de Sitter space (a
exp(H0t))

(31)

Comparing (30) and (31) we find
dLDS(z) >
dLMD(z),
which means that an object located
at a fixed redshift will appear brighter in
an Einstein-de Sitter universe than it will in de Sitter space
(equivalently in the steady state model).
(5)
This is also true for a
two component universe consisting of matter and a cosmological constant
as demonstrated in Fig 4. In a
spatially flat universe the presence of a
-term increases the
luminosity distance to a given redshift, leading to interesting
astrophysical
consequences. Since the physical volume associated with a unit redshift
interval increases in models with
> 0, the
likelihood that light from a quasar
will encounter a lensing galaxy is larger in such models. Consequently
the probability that a quasar is lensed by
intervening galaxies increases appreciably in a
dominated
universe, and can be used as a test to constrain the value of
[72,
71,
192].
Following
[73,
26,
37]
we give below the probability of a quasar
at redshift zs being lensed relative to the fiducial
Einstein-de Sitter model
(m = 1)

(32)

where d (z1, z2) is a
generalization of the angular distance
dA = dL(1 + z)-2
discussed in Section 4.5:

Turning now to the observational situation,
at the time of writing the best observational estimates give a
2 upper bound
< 0.66
obtained from multiple images of lensed quasars
[111,
112,
136].
Since radio sources are not plagued by some of the systematic
errors arising in an optical search (notably extinction in the lens
galaxy and the quasar discovery process) a search involving radio
selected lenses can yield useful complementary information to optical
searches [60].
Recent work by Falco et al (1998) gives
< 0.73 which
is only marginally consistent with optical estimates,
a combined analysis of optical and radio data yields a slightly more
conservative upper bound
< 0.62 at the
2 level (for flat
universes)
[60].
(Constraints on
from both
lensing and Type 1a Supernovae are discussed in
[201];
also see next section. An interesting new method of constraining
from weak
lensing in clusters is discussed in
[67],
also see
[12]
and section 4.6.)
Improved understanding of statistical and systematic uncertainties combined
with new surveys and better quality data promise to make gravitational
lensing a powerful technique for constraining cosmological parameters and
cosmological world models.

5 For instance a galaxy at
redshift z = 3 will appear 9 times brighter in a flat matter
dominated universe than it will in de Sitter space (see
Fig 4).
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